Einstein Hypersurfaces of \(\mathbb {S}^n \times \mathbb {R}\) and \(\mathbb {H}^n \times \mathbb {R}\)

Abstract

In this paper, we classify the Einstein hypersurfaces of \(\mathbb {S}^n \times \mathbb {R}\) and \(\mathbb {H}^n \times \mathbb {R}\). We use the characterization of the hypersurfaces of \(\mathbb {S}^n \times \mathbb {R}\) and \(\mathbb {H}^n \times \mathbb {R}\) whose tangent component of the unit vector field spanning the factor \(\mathbb {R}\) is a principal direction and the theory of isoparametric hypersurfaces of space forms to show that Einstein hypersurfaces of \(\mathbb {S}^n \times \mathbb {R}\) and \(\mathbb {H}^n \times \mathbb {R}\) must have constant sectional curvature.

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Acknowledgements

The authors would like to thank the referee for the valuable corrections and suggestions.

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Correspondence to João Paulo dos Santos.

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The third author was supported by FAPDF 0193.001346/2016.

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Leandro, B., Pina, R. & dos Santos, J.P. Einstein Hypersurfaces of \(\mathbb {S}^n \times \mathbb {R}\) and \(\mathbb {H}^n \times \mathbb {R}\). Bull Braz Math Soc, New Series (2020). https://doi.org/10.1007/s00574-020-00216-7

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Keywords

  • Hypersurfaces in product spaces
  • Einstein manifolds
  • Constant sectional curvature

Mathematics Subject Classification

  • 53B25
  • 53C40
  • 53C42